Theorems and Counterexamples in Mathematics

ISBN-13: 9781461269755 / Angielski / Miękka / 2012 / 305 str.

Bernard R. Gelbaum; John M. H. Olmsted

Opis

The gratifying response to Counterexamples in analysis (CEA) was followed, when the book went out of print, by expressions of dismay from those who were unable to acquire it. The connection of the present volume with CEA is clear, although the sights here are set higher. In the quarter-century since the appearance of CEA, mathematical education has taken some large steps reflected in both the undergraduate and graduate curricula. What was once taken as very new, remote, or arcane is now a well-established part of mathematical study and discourse. Consequently the approach here is designed to match the observed progress. The contents are intended to provide graduate and ad- vanced undergraduate students as well as the general mathematical public with a modern treatment of some theorems and examples that constitute a rounding out and elaboration of the standard parts of algebra, analysis, geometry, logic, probability, set theory, and topology. The items included are presented in the spirit of a conversation among mathematicians who know the language but are interested in some of the ramifications of the subjects with which they routinely deal. Although such an approach might be construed as demanding, there is an extensive GLOSSARY jlNDEX where all but the most familiar notions are clearly defined and explained. The object ofthe body of the text is more to enhance what the reader already knows than to review definitions and notations that have become part of every mathematician's working context.

Spis treści

1 Algebra.- 1.1 Group Theory.- 1.1.1 Axioms.- 1.1.2 Subgroups.- 1.1.3 Exact versus splitting sequences.- 1.1.4 The functional equation:f (x +y) = f (x) + f(y).- 1.1.5 Free groups; free topological groups.- 1.1.6 Finite simple groups.- 1.2 Algebras.- 1.2.1 Division algebras (“noncommutative fields”).- 1.2.2 General algebras.- 1.2.3 Miscellany.- 1.3 Linear Algebra.- 1.3.1 Finite-dimensional vector spaces.- 1.3.2 General vector spaces.- 1.3.3 Linear programming.- 2 Analysis.- 2.1 Classical Real Analysis.- 2.1.1 ?X.- 2.1.2 Derivatives and extrema.- 2.1.3 Convergence of sequences and series.- 2.1.4 ?X × Y.- 2.2 Measure Theory.- 2.2.1 Measurable and nonmeasurable sets.- 2.2.2 Measurable and nonmeasurable functions.- 2.2.3 Group-invariant measures.- 2.3 Topological Vector Spaces.- 2.3.1 Bases.- 2.3.2 Dual spaces and reflexivity.- 2.3.3 Special subsets of Banach spaces.- 2.3.4 Function spaces.- 2.4 Topological Algebras.- 2.4.1 Derivations.- 2.4.2 Semisimplicity.- 2.5 Differential Equations.- 2.5.1 Wronskians.- 2.5.2 Existence/uniqueness theorems.- 2.6 Complex Variable Theory.- 2.6.1 Morera’s theorem.- 2.6.2 Natural boundaries.- 2.6.3 Square roots.- 2.6.4 Uniform approximation.- 2.6.5 Rouché’s theorem.- 2.6.6 Bieberbach’s conjecture.- 3 Geometry/Topology.- 3.1 Euclidean Geometry.- 3.1.1 Axioms of Euclidean geometry.- 3.1.2 Topology of the Euclidean plane.- 3.2 Topological Spaces.- 3.2.1 Metric spaces.- 3.2.2 General topological spaces.- 3.3 Exotica in Differential Topology.- 4 Probability Theory.- 4.1 Independence.- 4.2 Stochastic Processes.- 4.3 Transition Matrices.- 5 Foundations.- 5.1 Logic.- 5.2 Set Theory.- Supplemental Bibliography.- Symbol List.- Glossary/Index.